This also happens in Magnetism, , and A is called magnetic vector potential
This is valid because the divergence of B is 0
it has gauge freedom that we can choose taht it is divergence freechoose
it looks like three independent Poisson equations
Example: If we have a one dimensional current i.e. wire, this current will produce
Which is Biot-Savart law!
Multipole expansion
Laplace with azimuthal symmetry:
where is Legendre polynomial
Apply to a point charge at
(unit charge)
This has azimuthal symmetry, so we can use the above equation
If is zero, then , then , then
Because (1)=1
The expansion would be
Choose the case
The general solutiuon would be
Thus the general solution for outside a charge distribution is
The leading term is the monopole term (l=0)
Which is proportional to
Next is the dipole term (l=1)
Which is proportional to
Where is the dipole moment
and
Then the quadrupole term (l=2)
Now we can have a nice way of looking at field outside the charges using multipole expansion.
2. Electric polarization
Matter is not free space, it contains atoms
The first two pictures are from Griffith Book
2.1 Bound charges
Electric field can induce a dipole moment in a neutral atom
And the electric moment is defined as where is the polarizability of the atom. A list of polarizability can be seen from the handout.
If, for example, the outside E ~ V/m, then the displacement would be about m, which is much less the size of an atom.
Also, there are polar molecules which already have a dipole moment, but they are randomized because of the present of inner energy
However, an E field can line them up
For a bok of aligned dipoles (per unit volume), define the polarization as the dipole moment per unit volume as
If they are not aligned, then the polarization is zero.
Question: what is the electric potential produced by a box of dipoles?
Note that
where is the surface bounded charge density and is the volume bound charge density (the minus sign is because is defined as the dipole moment per unit volume).
The derivation can be seen from Griffiths' book page 176.
Remember to drop primes!
In bulk, Gauss's Law is , but remember that and hence
We define electric displacement
That is much easier, since it only depends on free charges.
The integral form is
Linear dielectrics
For a linear dielectric, where is the electric susceptibility, which is dimensionless.
is the relative permitivity and it is dimensionless.
Example:
Field from a point charge in a linear dielectric:
Recap of lecture 5:Polarization: Bound charges: , Electric displacement: Maxwell: Linear dielectric: Where is the electric susceptibility =
Question Has D solved everything?-- No. The div equation is easier, but...
Reminder: And the equation for is
Boundrary conditions
is continuous
But is not continuous (if at interface)
is continuous
but may not be continuous if we have bound charges at the interface
Example
is continuous
So there is a smaller E field in the dielectric
Since is constant,
Example:
There is dielectric sphere, with radius a and relative permittiaty \epsilon_r. The sphere is in a uniform electric field . Find V.
We got to use Laplace's equation
General solution to Laplace's equation in spherical coordinates (with azimuthal symmetry) is
where is the Legendre polynomial.
Solve separatively for r > a and r < a
As , As , should be finite (not blow up) for r<a
case r<a:
And for r>a:
As for r = a, are continuous
is continuous
generates same expressions
That is not terribly useful, so we will try another one
is continuous
Hence, we got two expressions for A in terms of l. They cant both be right unless for
On the other hand, we have
①②
①②
Hence
Now, put everything together
Interestingly, we all have uniform fields applied inside and outside. And if in vacuum, we will have uniform field outside (dipole field vanishes)
P inside sphere:
where is
Dipole moment of sphere
Polarizability
Bound charge on the surface
3 Magnetization
3.1 Current Loop
From the handout, we have the following vector identity:
Start from Poisson's equation:
Current loop:
recall the expansion
l = 0 term is zero
n = 1 term is the next most important term
and hence
where
Is the magnetic dipole moment of the current loop.
Lets stop the expansion before we get too complex stuff.
And that looks like a dipole right?
Put in spherical coordinates:
which is same as electric dipole with
Field patterns look the same at large r, but at small r, the magnetic field is not singular.
3.2 Magnetic Properties
In a field ,a magnetic material will acquire a magnetization where n is the number of magnetic dipole and is the magnewtic moment of one atom
There are three main effects:
diamagnetism and that effect is very weak
All materials exihibits it!
Para-magnetism and it's stronger
That is often shown in materials with unpaired "spins"( but only at small B)Example:
Ferromagnetism
Very strong QM effect E.g. Fe, Co, NiIt is a non-linear effect of M(B)
Recap to lecture 7:
Current loop Magnet moment where S is the vector area of the loopMagnetization where n is the number of magnetic dipole and is the magnewtic moment of one atom
3.3 Field due to a box of magnetic dipoles
When doing this, we start with some vector identities:
start with Divergence theorem:
set where is a constant vector
LHS: because is a vector perpendicular to
RHS:
Therefore, for any ,
Assume there is a box of magnetic dipoles, and there is one dipole at position
which equals
We have
Hence
The last term can be transformed into a surface integral:
This looks like
Where is the bulk bound current density and is the surface bound current density.
Remember to drop the prime!
Having done this, we can fix up Amperes law:
Amperes law:
Where is the free current density where you can connect or do sth like that; and is the bound current density, it can be written as the curl of M
Hence the expression for free current would be
Stuff in the bracket is defined as the magnetic field H and hence the free current density has a simpler form
Its integral form is
However, becomes
The equtions to remember would be
Linear materials
M is proportional to Bwhere is the magnetic susceptibility
where is the relative permeability
If willing to be strict, H would be called magnetic field strength and B would be called magnetic flux density
3.4 Boundary conditions
one is not changed:
Making this cylinder flater and flater, we can see that the B field is continuous
However, is not true because the
(Assuming there is no surface free currents)
But is not continuous because you can have bound surface currents at interface and
Assuming there is no surface free charges/currents
3.5 Magnetic scalar potential
If everywhere, then
where is the magnetic scalar potential
If, we are dealing wit linear material,
If, in addition,
Then we can use the Laplace equation to solve for
3.6 Ferromagnetism
One way to define magnitization is = Microscopely, ferromagnet has M 0, even in B = 0The reason for ferromagnetism is the (what will be learnt next term), the exchange interactionOnly in certain materials:Fe, Co, Ni, Gd
There is an energy cost due to the stray field of the magnetic dipoles.
From the picture having stray field, it can be seen that on the tip and bottom of the field, there is a divergence of magnitization, leading to divergence in H. Meaning that field spreads out whenever you have magnetic moment going into the surface and not flowing out.
The energy cost would be
It is energetic flavorable to form domains
This will cost low energy since there is no stray field and hence no divergence in H. (Average over the surface, M is zero)
When applying a B field, the correlated domain is expanded and finally they are all aligned.
So, the magnetization process involves moving domain walls.
The process is highly non-linear hysterisis loopStates that is interesting
Saturation (right up)
Remanence (middle up)
Coercivity (left middle)Better picture is in handout
The field which is positive but M is zero is called coercive field.
We can modle this as
only if
and is a multivalued function (because it depends on histroy)
Hard materials
, are large
difficult to move domain walls
So it is hard to magnetize and demagnetize
so it is used for permanent magnets
Soft materials
, are small
easy to move domain walls
so it is used for transformers, motors, etc.
Example: Magnetization of a ring Iron ring, radius r, and current I with N terms used to magnetize the ring
Have a Ampere loop around the ring, then useSince H only depends on the free current, it is favorable for us to use it.
Continue the calculation
Hence B =
Now lets put a gap in the ring
The cutout, x, would much less than r
Lets use the same trick, ampere's law
Is continuous, Substitute in the function
Rearrange that
If , then becomes about
Is could be very large if x is small. But for that condition to work, you need to be very large.
4 Electromagnetic waves in material
Let's let everything move!
4.1 Displacement current
Conservation of charge
However, this is incompatible with
Lets take the divergence for both sides, we get
add an additional term to the current density
LHS:
Where is the bound current density.
RHS
we could write
where is the polarization current density. which equals
Note that from conservation of charge.
Thus the polarization current responds to changes to bound charge, and hence in
4.2 Maxwell's equations in insulating linear dielectrics
Since it is insulating linear dielectrics, we have and
Hence, we could get Maxwell's equation
remember, and
which gives
Consider
which is wave equation.
where c = and n = where n is also called refractive index
Plane waves solutions
Lets choose propagation parallel to z, and hence
remember that similarly,
we also have
And
Hence, and are constant in z and t, they are not part of wave motion
now analyze the x,y components of curl:
are solutions
Lets then take
Then, we could have
And then we could get the wave travelling in direction
Define Impedance Z as
remember that
The motivation of doing so is that and
So dimension would work
For free space, then, and
Remember that , and use E, B , we could get
Which gives the same answer because
which is this wave
4.3 conductors
Remember thatThe last one is 0 because
For conductors, we have
since there are no free charges in equilibrium from Ohm's law where is the conductivity and from linearity
Then we could get Maxwell's equation in conductors
Free charge will decay to zero in a short time , and it is easy to prove (said Blundell)
Where is equal to from Ohm's law
and is equal to from Gauss's law
Where
If the metal has great conductivity, then is very small, and hence is very small.
Let's consider the electromagnetic wave having frequency , so we would like to compare with :
Condition
Conductor Type
Charge Response
Conductivity
Good conductor
Charges respond very quickly
Conduction current dominates
Bad conductor
Charges respond very slowly
Displacement current dominates
Take real life examples
metal
1
Silicon
11.7
Glass
5
Note that visible light would have frequency ~ Hz
Let's now do some electromagnetism
Again, this would yield transverse plane waves with to each others
with
Taking the positive root
Sub into original equation
Where is the skin depth
Reminder: Good conductors have
We could therefore have
We could also neglect the last term, since
Hence, we could have
For a typical metal, is
Lets go to poor conductors
Poor conductors has , hence
which equals to which is independent to
In an insulating dielectric, , hence , and hence as expected.
Lets return to previous equation
Lets consider the curl equation in the conductor
If we expand, would be
For a good conductor, , hence
So this means that B lags behind E in a metal
4.4 Poynting vectors
Work done on charge
where eaquals to
Rate of work on charges
From Maxwell's equation:We have
By dotting everything, and then
Where we call "" (remember that which equals to energy stored in EM field per unit volume) and as "" or Poynting vector.
How is this working?
Assume that we are using a linear media:Remember that
Bringing everything together, we could get
Where is the poyting vector, or equiviantly,
We could say that, therefore is the energy flux density, or the rate of flow of energy per unit area in the direction of S.
Example: a capacitor
The stored energy increase at rate
also:
Hence, we have
where
There is another example
4.5 Radiation pressure
EM waves are made up of photons, and hecne they have momentum
Transport of energy is appoinated by transport of momentum
For a perfect absorber, where is the radiation pressure
Example For a plane EM wave in free space, we have
Where I is the intensity of wave
Sunlight: I~FYI,
Example Consider a star which is growing by accretion
i.e. matter is falling onto it uniformly in all directions
The star has luminosity L (e.g. w)
Energy flux =
Radiation pressure:
Outward:
Force/unit mass = where k is the opacity, which is area/unit mass, which is a constant
Inward: Force due to gravity/unit mass =
Since they balance, we have
which is called Eddington Limit (Upper limit of luminosity of stars that accrete (isotopically))
4.6 EM waves - reflection and refraction
Left:
Right:
Using electromagnetic boundary conditions, we could get
is continuous
is continuous
Putting two equations together
Where
|Poyting vector|
We expect
Where they equal to
separately
Lets now have angles
Choose in x-z plane
At z = 0, is continuous and this holds for all x y and t must be the same
for all x, y at z=0
Take
and all lie in the xz plane (the plane of incidence)
Take so
Remember that , And the last two would lead to
Where
Fresnel equations
Worring about polarization directions
We work in those steps
in the plane of incidence
"parallel-like" = parallel
Remind that and form a right-handed system
incident
reflected
transmitted
continuous continuous
Now look for Fresnel equations for p-polarizations
perpendicular to the plane of incidence
"s-like" s = senkrecht = perpendicular
incident
reflected
transmitted
continuous continuous
Remember that
Let's set Then,
So we can replace with in expressions involving ratios of Z's.
e.g. Fresnel equations for p-polarization
Use Snell's law
For s-polarization, we have
We also have
Fresnel equations:
on top and bottom
(p)
(s)
Remember, EM waves have an energy flux given by
Intensity coefficients
where is due to waves at ifferent speeds. And is due to the wavesfronts at different angles.
Lets check in certain cases
Example: air/glass interface
4% of light is reflected, 96% is transmitted
[If , for example, = 1.75, , which is sa problem]
Lets take another go at differnet angle
, and set
now consider , we cna have total interal reflection for where is the critical angle =
At ,
Hence, we have
vanishes at the certain angle called "Brewster's angle" when
Method of producing polarized light
Reflectted light polarized with perpendicular to the plane of incidence
Polarizing sunglasses with transmission axis vertical reduce glare because reflected light is mainly horizontally polarized.
Total internal reflection
Since , there will be total internal reflection when
For transmitted wave, its like
in medium= since = where This is called an evanescent wave
Lets now consider the Plane travelling wave
For s-polarization, , we have
So is complex
where
is in phase with transport of energy along x is out of phase with no transport of energy along z is the complex conjugate of H
Reminder on conductors
Reflection from a meatal surface
air
metal
for a good conductor because
write
where is the skin depth most of the EM wave intensity is reflected metals are shiny!
4.7 Plasmas
Plasmas are neutral gas of charged particles, such as ions and free electrons (like metals)
Examples: where you can find plasma
metal
ionosphere
stars
fusion reactor
interstellar gas /intergalactic medium
supernova remnants
radio galaxies / quasars
lightening
aurorae
fire
plasma displays
Its density varies from , temperature is
we would only focus on cold plasma
Lets consider a slab of plasma
having number density n
the positive ions are fixed in place, and now lets move electrons by distance
E field: and
This is the SHM at the plasma frequency
Now, lets drive charges with EM wave (we could ignore B if )
Hence, we could get
where is the refractive index, and that can be imaginary
we could plot the relation between and
at , is real, hence EM waves can propagate
at , is imaginary, hence EM waves cant propagate
For example, metals are shiny, but only at optical frequencies. They will transport if it is going to much shorter wavelengths.
e.g. ionosphere
AM radios would be refracted + reflected (~ 1 MHz)
FM radio and TV radios would escape (~100MHz)
Lets look at the dispersion relation again
we could get
The dispersion relationship would be like
Hence, we could conclude that
waves are dispersive in plasma
there are no propagating waves for
waves with are slow as
We have
We could have end behaviors:
Let's then take
Choose to be , we have
We could hence get Maxwell's equation
For the wave, we have
Transverse solutions: we have
As for longitudinal solutions, we have , we have, hence,
We can therefore classify waves in the k graph
4.8 Dispersion
Refractive index changes with frequency
classical theory of dispersion:
model electrons as a classical damped oscillator
Assume that
As , , As , , we could get a better illustration, therefore, for the real part and imaginary part of
We can also conclude that corresponds to the absorption of light, and corresponds to the refraction of light and generally increases with frequency
The sharp drop in the real part of in the real part of near is called anomalous dispersion
5. Confined EM waves
5.1 Transmission lines
An example of guided wave
Lets think in a super long circuit
If the time for signal to transverse the circuit is not , we need to consider the wave behavior of the signal.
We could have
Remember: we are defining capacitance as C per unit length, and inductance L per unit length
If we plug equation 1 into equation 2, we get
For wavelike situations, we have
where and are arbitrary functions, and is the wave velocity which is .
Note: the wave velocity is not the speed of light, but the speed of the wave in the circuit.
From equation 1, wee could have The impedance would be, then where thr is the direction of the wave.
Instantaneous power
Example
We have coaxial transmission line, which has
Remember: is per unit length
We would have
Hence, we have
Where, remember, is the flux per unit length
Then, a common type of transmission line:
Example: Strip transmission line
so we could ignore edge effect
We have
Boundary between two transmission lines
For x<0, we have
For x>0, we have
At x = 0, we need to match voltages
Match currents, we have
From (1) and (2), we have ,
As for power, we would have as expected.
Termination of load
is the boundary condition
We would also like to consider special cases
Short circuit
0
-1
0
0
Open circuit
1
2
0
Matched
0
1
When , we would get maximum power transfer
proof
Incident power
Power on the load Hence,
Input impedance of short sections
Input impedance
If = 0 If =
For a line, we have
Choose i.e. The transmission line has been perfectly terninated and there is no reflected wave
Waveguides
Example of interfering EM waves
Consider 2 EM waves with , travelling along
And we add them up
And so it is zero when i.e. when a
Nodal plane would be like
We could, therefore, insert conducting sheets at places where
This demonstrates (at least in 1 dimension) that guided waves are possible
When confined
only is transverse, is not
Rectangular waveguide
Rectangular cross section metal walls
, inside conductor walls and are continuous and are zero at the walls
Consider Maxwell's equations inside waveguide
Hence, using our old method, we could haveWhere the LHS would be , so we can have a second order differential equation
Which is called the Helmholtz equation
One set of solution of this equation have transverse (called TE modes)Since = 0 at walls, we could get And since , we would like to put it in Helmholtz equation
Notice that = 0
Boundary conditions
From , we have , hence we could continue
If m = 1, n = 0, we would call it TE10 mode
We could have Helmholtz equation for as wellWhere is the cutoff wave vector
There is a minumym frequency allowed allowed by the waveguide
And
We can, therefore, consider the mode ad an EM wave bouncing off the walls at angle with respect to the walls
6. Special relativity and electromagnetism
Just a brief introduction, no worries.
We have spacetime four vector (), who has dot product with itself as
We also have momentum four vector (), who has dot product with itself as
We also have differential operator which is written as
We would just introduce this here
Then, we could have current density four vector
We would consider charge at rest, then
This could be "boosted" to a moving frame with speed v
Where due to length contraction and
Then, we could have continuity equation
In lab frame S, then, we could have
Line of charge, with density which is stationary
Line of charge, with density which is moving with speed
Test charge q moving with speed
remember that the wire itself has no net charge
In the test charge frame S', we could have
Test charge is stationary
+ charges move backwards at speed u
- charges move back at
Hence
In the rest frame of negative charges, they have charge density of
Lets remember Maxwell's equations
We would, therefore, need a new object called the field strength tensor
And the function gives us the four Maxwell's equations. The first half would be simple to deduce, but the second half would be a bit more complicated so we are not going to do it here.
First two:
Bonus Lecture: Signals and sampling
Not in the exam
Fourier transform is a way to represent a signal as a sum of sinusoids. It is a continuous function of frequency.
This is just a convention, and the latter would be what we use in this course or lecture.
A signal has a frequency spetrum .
From tis, we could say thatthe signal is band limited if for .
And we could get some examples of band limited signals from the lecture.
Look at some examples of fourier transform.
Delta function
That would give a unniform frequency spectrum, which is not band limited.
If you fourier transfor a delta function which is not at the origin, you would get a complex exponential.
We will now think of something called the Dirac comb or sampling function.
The idea of the dirac comb is that we would have a fourest of delta functions, which are separated by Ts, and we would have a delta function at every integer multiple of .
This is a period function with period .
WE could, therefore, write its fourier transform as a sum of fourier transforms of delta functions.
Where would be
This looks difficult, but since is a sum of delta functions, and only one would be nonzero in the region of integration, we would get
Now, we woule have fourier transform of the entire Dirac comb.
If you compare the two things we have, we would see that they are the same, and we would get
In omega space, the delta functions are separated by .
We have seen that fourier transform of a delta function is a constant, and we would have a constant at every integer multiple of .
Go back to problem of sample a function , at a frequency Consider
Since this is a product of two functions, we could write its fourier transform as a convolution of the fourier transforms of the two functions.
Maybe just look at how this work, if we take the fourier transform of , we would get
Where is a function taht can be written as
Now put everything together, we get
The integration of dirac comb is just the sum of the integrations of the delta functions, which can be written as
And hence
That is just the convolution theorem tho.
If we look at the fourier transform, we get lots of copies of the , and they are separated by .
This idea of having multiple copies of the fourier transform of the original function is important
If you think of samping a function, lets say a sine wave, you digitize it is certain points
The question is how few points do you need to digitize a signal
The idea would be , where is the frequency of the signal.
You could, for example, sample from peak to trough, and you would get a good approximation of the signal, which should be the max frequency of sampling
This is known as the Nyquist frequency.
If we go back to the particular case of this signal, which is band limitecd
They would be nicely separated, not touching each others
BUT if , then the copies would overlap, and you would not be able to tell which is which.
This is called aliasing.
This is a problem, decreasing its quality, and we would like to avoid it.
To avoid it the Nyquist frequency should be twice as big as the maximum frequency of the signal.
Lastly, lets talk about Johnson-Nyquist noise
Everything has noise, everything is fluctuating
This circuit actually has normal modes, , and since , one mode in a frequency interval of .
We also have
This is a result of the equipartition theorem.
Hence
That is just for one mode, in other word, in a ferquency unit
S, the mean power on the resistor on the right
which equals to
which is named powwer
we have , and hence, which is known as Johnson-Nyquist noise across R in a frequency interval .